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Asymptotic Behavior of a Viscous Liquid-Gas Model with Mass-Dependent Viscosity and Vacuum

In this paper, we consider two classes of free boundary value problems of a viscous two-phase liquid-gas model relevant to the flow in wells and pipelines with mass-dependent viscosity coefficient. The liquid is treated as an incompressible fluid whereas the gas is assumed to be polytropic. We obtain the asymptotic behavior and decay rates of the mass functions $n(x,t)$,\$m(x,t)$ when the initial masses are assumed to be connected to vacuum both discontinuously and continuously, which improves the corresponding result about Navier-Stokes equations in \cite{Zhu}.